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3D BÉZIER CURVE

Link to a figure manipulable by mouse


Curve studied by Bézier in 1954 and by de Casteljau.
Pierre Bézier (1910 - 1999): engineer for Régie Renault.

 
Affine parametrization:   (i.e. ) where the  are the Bernstein polynomials: .
Polynomial algebraic curve of degree n.
Curvature at A0 , torsion at A0 .

 
Given a broken line  (called control polygon, the Ak being the control points), the associated Bézier curve is the curve with the above parametrization; the curve goes through A0 (for t = 0) and An (for t = 1), and the portion that links these points is traced inside the convex hull of the control points; the tangent at A0 is (A0A1) and the tangent at An is (An-1An).
Animation of the evolution of a cubic Bézier curve with 4 control points, where A1 and A4 are fixed, A2 and A3 move on lines.

Recursive construction (de Casteljau algorithm):
The point is the barycenter of  and  where  are the respective current points of the Bézier curves with control points  and ; moreover, the line  is the tangent at  to the Bézier curve.

Conversely, any polynomial 3D algebraic curve is a Bézier curve associated to a unique polygon, once the vertices of the polygon are chosen arbitrarily on the curve.

Their planar conical projections are the plane rational Bézier curves.
 
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© Robert FERRÉOL 2018